Alexandrov space

Builds on Metric space Complete metric space Locally compact space CAT(k) space

Definition

Let $k$ be a real number. Let $M_k$ and $D_k$ be the model surface of constant curvature $k$ and its diameter bound, as for CAT(k) spaces. Let $(X, d)$ be a metric space. The space $X$ is a length space if each distance $d(x, y)$ is the infimum of the lengths of continuous curves from $x$ to $y$ . Geodesic triangles and comparison triangles in $M_k$ are as for CAT( $k$ ) spaces.

The space $X$ is an Alexandrov space of curvature $\ge k$ if:

$X$ is a complete, locally compact length space, and
each point of $X$ has a neighbourhood $U$ with the following property. Take any geodesic triangle in $U$ of perimeter less than $2 D_k$ , any vertex $x$ and any point $p$ on the opposite side. Let $\bar x, \bar p$ be the comparison points in a comparison triangle in $M_k$ . Then $d(x, p) \ge d_{M_k}(\bar x, \bar p)$ .

Many authors also require $X$ to have finite Hausdorff dimension; that dimension is then an integer.

Remarks

Example: a complete Riemannian manifold with sectional curvature $\ge k$ is an Alexandrov space of curvature $\ge k$ , by Toponogov's theorem. In particular $\R^n$ has curvature $\ge 0$ ; its triangles equal their comparison triangles.

Non-example: the hyperbolic plane of curvature $-1$ does not have curvature $\ge 0$ . Take a geodesic triangle in it and the midpoint of one side. The distance from the opposite vertex to this midpoint is strictly less than in the flat comparison triangle, so clause 2 fails for $k = 0$ .

For an integer $m \ge 2$ , let $C_m$ be the quotient of $\R^2$ by the group of rotations about $0$ through multiples of $2\pi/m$ . This cone is an Alexandrov space of curvature $\ge 0$ , and it is not a smooth Riemannian manifold at its tip.

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Tag	`QAAV27WAAP`
URI	`m2://QAAV27WAAP`
Kind	definition
Writing	W0 stub
MSC	53C23, 53C20
Defines	Alexandrov space
Uses	Metric space, Complete metric space, Locally compact space, CAT(k) space
Changed	2 November 2026

Alexandrov space

Definition

Remarks

Reading path

Further reading

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